I had the ceilings of my house, which contained asbestos removed by breaking them and letting them fall on the floor 7 years ago. My children keep falling sick every now and then. Could this be attributed to the asbestos dust that could have found its way into crevices and roof timbers?

Exposure to asbestos has been linked to the development of serious lung diseases, among them cancer and asbestosis (asbestosis is more linked to heavy, occupational exposure). What is common to all asbestos related diseases is a long latency period, i.e. disease only manifest years after being exposed (typically 15-50 years) following exposure. It is therefore, very unlikely that your children falling ill, is in anyway asbestos related.